Xenia is ordering a cake for her friend’s wedding. She knows that there will be 110 guests at the wedding, and she is trying to decide how large a cake to order. She has been told that each guest should get a piece of cake that is 4 inches by 4 inches. She would like to buy a tiered, square three-layer cake and is trying to decide which sizes of cakes to order. Which combination of cakes should she get? The available sizes, in inches, are:
32 × 32
28 × 28
24 × 24
20 × 20
16 × 16
12 × 12
8 × 8
Since 110 = 49+36+25, use the
28,24,20 inch cakes
To determine the combination of cakes that Xenia should order, we need to calculate the amount of cake needed to serve all 110 guests.
First, let's calculate the area of each cake size:
- 32 × 32 = 1,024 square inches
- 28 × 28 = 784 square inches
- 24 × 24 = 576 square inches
- 20 × 20 = 400 square inches
- 16 × 16 = 256 square inches
- 12 × 12 = 144 square inches
- 8 × 8 = 64 square inches
Next, we need to determine how many pieces of cake can be cut from each size of cake. Since each guest should have a 4 × 4 inch piece, we can divide the area of each cake size by the area of a single piece (4 × 4 = 16 square inches).
- 32 × 32: 1,024 / 16 = 64 pieces
- 28 × 28: 784 / 16 = 49 pieces
- 24 × 24: 576 / 16 = 36 pieces
- 20 × 20: 400 / 16 = 25 pieces
- 16 × 16: 256 / 16 = 16 pieces
- 12 × 12: 144 / 16 = 9 pieces
- 8 × 8: 64 / 16 = 4 pieces
Now, let's find the cake combination that can accommodate all 110 guests. We can start by considering the largest cake sizes and working our way down:
- 32 × 32: 2 cakes -> 2 × 64 = 128 pieces (exceeds the number of guests)
Since 2 cakes of the largest size exceed the number of guests, we can move to the next size:
- 28 × 28: 2 cakes -> 2 × 49 = 98 pieces (insufficient for 110 guests)
- 24 × 24: 3 cakes -> 3 × 36 = 108 pieces (sufficient for 110 guests)
Therefore, Xenia should order 3 cakes of size 24 × 24 inches. This combination will provide enough cake to serve all 110 guests.